Is there any specific data on this?
Nope.
I have seen a quality fluid run cooler, based on looking at an actual transmission temperature gauge, in a golf course mower. However, the Permatran is a quality fluid. And, Permatran3 was the actual fluid used in that test.
I am a fan of the Amsoil ATH. And, I have had it in my GC for over a decade.
People make lots of claims about it, which Amsoil does not make.
On paper it's not magic, unless you are operating in extreme temperatures. Or, want to extend your intervals.
Because of the long warm up times recommended in the GC owners manual for operation at very cold temps, I chose to use the synthetic, in case there was any edge I could gain there. But, If you look at the specifications, even the old Permatran 3, stacked up pretty well against the Amsoil ATH.
The OEM fluids are, finally catching up. And, they are all going to synthetic now too.
Lots of GC owners are using the Amsoil ATH. But, you would also be fine with the new Permatran.